Exercise 1.

Show that $R^∗ = R \ \left\{0\right\}$ is a group under the operation of multiplication.

We have two non-zero numbers: a, b

If $a, b \in R^* \rightarrow R^*$ is closed under multiplication.

Also, we have

• $G_1$ - associative: real number multiplication is always association

$$ \forall a, b, c \in R^* \Leftrightarrow (a * b) * c = a * (b * c) $$

• $G_2$ - identity element:

  If $a = 1 \Leftrightarrow a * 1 = 1 * a = a \forall a \in R^*$

• $G_3$ - inverse element:

  If $a \in R^* \Leftrightarrow \frac{1}{a} \in R^*$ and $a * (\frac{1}{a}) = 1$

$\Rightarrow$ $\frac{1}{a}$ is inverse element

Conclusion: All conditions are satisfied so $R^*$ is a group

Exercise 2

Given the groups $R^∗$ and Z, let $G = R^∗ × Z$. Define a binary operation ◦ on G by $(a, m) \circ (b, n) = (ab, m + n)$. Show that G is a group under this operation.

$G = R * Z$ and the binary operation defined as:

$$ (a, m) \circ (b, n) = (ab, m + n) $$

+) where $(a, m), (b, n) \in \R \times \Z$

+) Since $(1, 0) \in \R \times \Z$ ⇒ G is non empty

• Associative: let $(a, m), (b, n), (c, p) \in \R \times \Z$

$(a, m) \times ((b, n) \times (c, p))$